Factoring an integer with three oscillators and a qubit

A common starting point of traditional quantum algorithm design is the notion of a universal quantum computer with a scalable number of qubits. This convenient abstraction mirrors classical computations manipulating finite sets of symbols, and allows for a device-independent development of algorithmic primitives. Here we advocate an alternative approach centered on the physical setup and the associated set of natively available operations. We show that these can be leveraged to great benefit by sidestepping the standard approach of reasoning about computation in terms of individual qubits. As an example, we consider hybrid qubit-oscillator systems with linear optics operations augmented by certain qubit-controlled Gaussian unitaries. The continuous-variable (CV) Fourier transform has a native realization in such systems in the form of homodyne momentum measurements. We show that this fact can be put to algorithmic use. Specifically, we give a polynomial-time quantum algorithm in this setup which finds a factor of an $n$-bit integer $N$. Unlike Shor's algorithm, or CV implementations thereof based on qubit-to-oscillator encodings, our algorithm relies on the CV (rather than discrete) Fourier transform. The physical system used is independent of the number $N$ to be factored: It consists of a single qubit and three oscillators only.